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A002831
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Number of 3-edge-colored connected trivalent graphs with 2n nodes.
(Formerly M3424 N1388)
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20
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1, 4, 11, 60, 318, 2806, 29359, 396196, 6231794, 112137138, 2249479114, 49691965745, 1197158348160, 31230408793660, 876971159096883, 26374570956403684, 845812191249484022, 28812214090645864661, 1038982259432805270094, 39540452134474760212909
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OFFSET
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1,2
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COMMENTS
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In a letter to N. J. A. Sloane dated Feb 04 1971 (see link), R. C. Read enclosed a table listing 14 sequences, all of which, he says, appeared in his 1958 Ph.D. thesis. The values he gave for terms a(5) and a(6) in the present sequence are apparently incorrect (the terms given here are correct; the incorrect terms are shown in A246598). - N. J. A. Sloane, Sep 08 2014
Comment from Max Alekseyev, Sep 09 2014: the relationship between "all graphs" and "connected graphs" is of course a version of the Euler transform - see for example the third formula in the Euler Transform link.
From Sasha Kolpakov, Dec 17 2017: (Start)
Number of oriented unrooted pavings (after Arques & Koch, Spehner, Lienhardt) with 2n darts.
Also the number of conjugacy classes of free index 2n subgroups in the free product Z_2*Z_2*Z_2. (End)
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REFERENCES
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R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Andrew Howroyd, Table of n, a(n) for n = 1..30
L. Ciobanu and A. Kolpakov, Three-dimensional maps and subgroup growth, arXiv:1712.01418 [math.GR], 2017.
R. C. Read, Letter to N. J. A. Sloane, Feb 04 1971
Neriman Tokcan, Jonathan Gryak, Kayvan Najarian, Harm Derksen, Algebraic Methods for Tensor Data, arXiv:2005.12988 [math.RT], 2020.
Eric Weisstein's World of Mathematics, Euler Transform
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FORMULA
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G.f.: sum(mobius(k) * log(G(x^k)) / k, k >= 1) where G(x) is the g.f. for A002830. - Sean A. Irvine, Sep 09 2014
Asymptotics: a(n) ~ (2/Pi)^(1/2)*(2/e)^n*n^{n - 1/2}; cf. Ciobanu and Kolpakov in Links. - Sasha Kolpakov, Dec 17 2017
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MATHEMATICA
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terms = 20;
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t k; s += t]; s!/m];
b[k_, q_] := If[OddQ[q], If[OddQ[k], 0, j = k/2; q^j (2 j)!/(j! 2^j)], Sum[ Binomial[k, 2 j] q^j (2 j)!/(j! 2^j), {j, 0, Quotient[k, 2]}]];
pm[v_] := Module[{p = Total[x^v]}, Product[ b[Coefficient[p, x, i], i], {i, 1, Exponent[p, x]}]];
a2830[n_] := Module[{s = 0}, Do[ s += permcount[p] pm[p]^3, {p, IntegerPartitions[2 n]}]; s/(2 n)!];
G[x_] = 1 + Sum[a2830[n] x^n, {n, 1, terms+1}];
gf = Sum[MoebiusMu[k] Log[G[x^k]]/k, {k, 1, terms+1}] + O[x]^(terms+1);
CoefficientList[gf, x] // Rest (* Jean-François Alcover, Jul 02 2018, after Andrew Howroyd *)
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CROSSREFS
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Cf. A002830 (for not-necessarily connected graphs), A246598 (for the incorrect values). See also A006712, A006713.
Sequence in context: A203577 A081073 A245545 * A246598 A242749 A303955
Adjacent sequences: A002828 A002829 A002830 * A002832 A002833 A002834
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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a(5) and a(6) corrected and new terms a(7) and a(8) computed by Sean A. Irvine, Sep 09 2014
a(9)-a(10) from Sasha Kolpakov, Dec 11 2017
a(11) and beyond from Andrew Howroyd, Dec 14 2017
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STATUS
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approved
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